Calculus of variations

“ The simplest problem in the calculus of variations proper is known to consist in finding a function of a variable such that the definite integral assumes a minimum value as compared with the values it takes when is replaced by other functions of with the same initial and final values. ”

“ But in the mathematical or pure sciences,—geometry, arithmetic, algebra, trigonometry, the calculus of variations or of curves,—we know at least that there is not, and cannot be, error in our first principles, and we may therefore fasten our whole attention upon the processes. As mere exercises in logic, therefore, these sciences, based as they all are on primary truths relating to space and number, have always been supposed to furnish the most exact discipline. ”

“ The formation of the first variation of an integral by means of a weak variation can be carried out without difficulty in the case of a simple integral involving any number of dependent variables and differential coefficients of arbitrarily high orders, and also in the cases of double and multiple integrals ”

“ This equation, or equations, will be referred to as the principal equation, or principal equations, of the problem.
The problems and method of the calculus admit of more exact formulation as follows: We confine our attention to the case where the sought curve is plane, and the function F contains no differential coefficients of order higher than the first. Then the problem is to determine a curve joining two fixed points (x0, y0) and (x1, y1) so that the line integral
taken along the curve may be a maximum or a minimum. ”

“ This can be done by expressing the value of the integral taken along a stationary curve from the point (x0, y0) to the point (x1, y1) in terms of the co-ordinates x1, y1, and then making this expression an extremum, in regard to variations ofx1, y1, by the methods of the differential calculus, subjecting (x1, y1) to the condition of moving on the curve C1.
An important example of the first variation of integrals is afforded by the Principle of Least Action in dynamics. The kinetic energy T is a homogeneous function of the second degree in the differential coefficients q1, q2, . . . ”

“ Thus, for example, in the case of rectifications, it will be first necessary to calculate dy, or dy and dz, by differentiating the equation or the equations of each curve proposed; after eliminating these expressions, the general differential formula above enunciated will then contain only ds and dx; having arrived at this point, the elimination of the infinitesimals can be completed only by the integral calculus. ”

“ This question is now resolved in the most complete and the most simple manner, as are all those of which the differential calculus is composed. It is easy to conceive the general importance which it must have in any of the applications of the transcendental analysis, the fundamental resources of which it may be considered as augmenting, by permitting us to choose (in order to form the differential equations, in the first place, with more ease) that system of independent variables which may appear to be the most advantageous, although it is not to be finally retained. ”

“ But when we come to the right-hand side of the equation we must remember that what we have got to sum up together is not all the ’s, but all such terms as ; and this will not be the same as , because is not a constant. For some of the ’s will be multiplied by big values of , and some will be multiplied by small values of , according to what happens to be. So we must bethink ourselves as to what we know about this process of integration being the reverse of differentiation. ”

“ By triumphs must be understood the successes with which the calculus has been applied to the solution of problems otherwise intractable. Often in the consideration of physical relations one is able to build up an expression for the law governing the interaction of the parts or of the forces that govern them, such expression being naturally in the form of a differential equation, that is an equation containing differential coefficients with or without other algebraic quantities. ”

“ If, indeed, the differential calculus was not rigorously complete, that is, if we did not know how to differentiate at will any function whatever, the differentiation to the second or higher order of each determinate function might engender special difficulties. ”

“ From the considerations indicated in this chapter, we see that, while the differential calculus constitutes by its nature a limited and perfect system, to which nothing essential remains to be added, the integral calculus, or the simple system of integration, presents necessarily an inexhaustible field for the activity of the human mind, independently of the indefinite applications of which the transcendental analysis is evidently susceptible. ”

“ They are those in which the differential equations are found to be immediately ready for integration, because they contain, at their first formation, only the infinitesimals which relate to the functions sought, or to the really independent variables, without its being necessary to introduce, differentially, other functions as intermediaries. If in these new cases we introduce these last functions, since, by hypothesis, they will enter directly and not by their differentials, ordinary algebra will suffice to eliminate them, and to bring the question to depend on only the integral calculus. ”

“ In the differential and integral calculus it is often advantageous to regard as multiple quantities various sets of variables, especially the independent variables, or those which may be taken as such. ”

“ Besides the method of finding the integrals of given differentials, the integral calculus is also applied to various branches of mathematics, as well as to almost every branch of natural philosophy and engineering. ”

“ The Differential Calculus, however, shows that the mind is feeling after even a more definite mode of contrast than that indicated above. We cannot truly know the Law which governs an organic group of phenomena, unless we allow ourselves to conceive the idea of a Singular Solution of that Law, i.e., an individual which is constant in some relations as to which the other individuals are variable, and variable in some relations as to which the others are constant. ”

“ And when such a differential equation has been found, one can get no further until it has been integrated. Generally it is much easier to state the appropriate differential equation than to solve it:–the real trouble begins then only when one wants to integrate, unless indeed the equation is seen to possess some standard form of which the integral is known, and then the triumph is easy. ”

“ Infinitesimals. In the Differential Calculus we are usually concerned with the derivative, that is, with the ratio of the differentials dy and dx. In some applications it is also useful to consider dx as an infinitesimal (see §15) , that is, as a variable whose values remain numerically small, and which, at some stage of the investigation, approaches the limit zero. ”

“ A direct and an inverse calculus is thus created, the object of the former being to determine the coefficients from the generating function, of the latter to discover the generating function from the coefficients. The one is a problem of interpolation, the other a step towards the solution of an equation in finite differences. ”

“ Since the formulas deduced in the Differential Calculus generally involve derivatives of y with respect to x, such formulas as (A) and (B) are especially useful when the parametric equations of a curve are given. ”

“ Such men seem to forget the great lesson that the differential calculus has taught us. It represents one of the greatest developments of modern mathematics. It does not solve problems by absolute solutions but by such approximations as make the answer which cannot be reached very clear. ”

“ Since the differential of a function is its derivative multiplied by the differential of the independent variable, it follows at once that the formulas for finding differentials are the same as those for finding derivatives given in § 33, if we multiply each one by dx. ”

“ The one possesses the highest possible degree of generality; for all things whatever admit of enumeration. The others are less general; seeing that there are endless phenomena that are not cognisable either by general geometry or rational mechanics. In conformity with the alleged law, therefore, the evolution of the calculus must throughout have preceded the evolution of the concrete sub-sciences. ”

“ The derivative of the quotient of a function by a constant is equal to the derivative of the function divided by the constant.
All explicit algebraic functions of one independent variable may be differentiated by following the rules we have deduced so far. ”

“ Examples of analytical expressions which represent continuous functions that are not differentiable have been given by Riemann, Weierstrass, Darboux and Dini (see § 24) . The most important theorem in regard to differentiable functions is the “theorem of intermediate value.” (See Infinitesimal Calculus.)
13. Analytic Function.—If ƒ (x) and its first n differential coefficients, denoted by ƒ′ (x) , ƒ′′ (x) , ... ƒ (n) (x) , are continuous in the interval between a and a + h, then
where Rn may have various forms, some of which are given in the article Infinitesimal Calculus. ”

“ The limit expressed by the right-hand member of this defining equation is often written
and is called the “derived function” of ƒ (x) , sometimes the “derivative” or “derivate” of ƒ (x) . When the function ƒ (x) is a rational integral function, the division by Δx can be performed, and the limit is found by substituting zero for Δx in the quotient. For example, if ƒ (x) = x2, we have
and
The process of forming the derived function of a given function is called differentiation. ”

“ The fundamental artifice of the calculus is the artifice of forming differentials without first forming differential coefficients. From an equation containing x and y we can deduce a new equation, containing also Δx and Δy, by substituting x+Δx for x and y+Δy for y. If there is a differential coefficient Fundamental Artifice.of y with respect to x, then Δy can be expressed in the form φ.Δx + R, where lim.Δx = 0 (R/Δx) = 0, as in § 7 above. ”

“ It is even certain that the last special improvements in mathematical analysis have contributed to extend our resources in that respect, by introducing within the domain of the calculus certain definite integrals, which in some respects supply the place of new simple functions, although they are far from fulfilling all the necessary conditions, which has prevented me from inserting them in the table of true analytical elements. ”

“ And, indeed, it is the distinctive peculiarity of the calculus in general, that it proceeds on the hypothesis of that absolute equality of its abstract units, which no real units possess; and that the exactness of its results holds only in virtue of this hypothesis. The first ideas of number must necessarily then have been derived from like or equal magnitudes as seen chiefly in organic objects; and as the like magnitudes most frequently observed were magnitudes of extension, it follows that geometry and arithmetic had a simultaneous origin. ”

“ In the first place, the possibility of drawing them at all, manifestly supposes that we are acquainted not only with the variations, but with the absolute quantities both of A and a. If we do not know the total quantities, we can not, of course, determine the real numerical relation according to which those quantities vary. ”

“ This transformation can always be effected. On the other hand, every differential equation of any order in relation to a single function might evidently always be reduced to the first order, by introducing a suitable number of auxiliary differential equations, containing at the same time the different anterior derivatives regarded as new functions to be determined. ”

“ As for subtraction, it follows at once; for if the function had itself had a negative sign, its differential coefficient would also be negative; so that by differentiating
,
we should get
.
But when we come to do with Products, the thing is not quite so simple.
Suppose we were asked to differentiate the expression
,
what are we to do? The result will certainly not be ; for it is easy to see that neither , nor , would have been taken into that product. ”